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Confidence Interval Calculator
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Confidence Interval Calculator is used to calculate the confidence limits for mean with respect to the given values of sample size, standard deviation, mean and confidence level.

### Confidence Interval Formula

If $(n\ \geq\ 30)$

Confidence Interval = $x\ \pm$ $z_{\frac{\alpha}{2}}$ $\times$ $\left(\frac{\sigma}{\sqrt{n}}\right)$

If $(n < 30)$

Confidence Interval = $x\ \pm$ $t_{\frac{\alpha}{2}}$ $\times$ $\left(\frac{\sigma}{\sqrt{n}}\right)$

Where,

$x$ = Sample Mean

$\sigma$ = Standard Deviation

$\alpha$ = $1$ - $\left(\frac{Confidence\ Level}{100}\right)$

$Z_{\frac{\alpha}{2}}$ = Value of the z-table

$t_{\frac{\alpha}{2}}$ = Value of the t-table.
You can see a default confidence level, sample size, standard deviation, mean given below. Click on "Calculate". Confidence interval is calculated by applying the default values in the confidence interval formula.

## How to Calculate Confidence Interval

Step 1 :

Observe the value of given mean and standard deviation for a respective sample size to find the confidence interval at a particular confidence level for mean percent.

Step 2 :

Apply the Confidence Interval Formula:

If(n $\geq$ 30)
Confidence Interval = x $\pm$ z$_{\frac{\alpha}{2}}$ $\times$ $\left(\frac{\sigma}{\sqrt{n}}\right)$
If(n < 30)
Confidence Interval = x $\pm$ t$_{\frac{\alpha}{2}}$ $\times$ $\left(\frac{\sigma}{\sqrt{n}}\right)$
Where,
x = Sample Mean
$\sigma$ = Standard Deviation
$\alpha$ = 1 - $\left(\frac{Confidence\ Level}{100}\right)$
Z$_{\frac{\alpha}{2}}$ = Value of the z-table
t$_{\frac{\alpha}{2}}$ = Value of the t-table.

## Confidence Interval Example

1. ### In a survey of 20 persons to find what percent of their income is given to charity, discover mean percent is 16 with a standard deviation of 6 percent. Find the confidence interval of mean percent at 95%?

Step 1 :

Given Sample size = 20 persons

Sample mean = 16

Standard deviation = 6

Confidence interval of mean percent = 95% = 0.95

Step 2 :

Since n < 30, So
Confidence Interval = x $\pm$ t$_{\frac{\alpha}{2}}$ $\times$ $\left(\frac{\sigma}{\sqrt{n}}\right)$
Confidence Interval = 16 $\pm$ t$_{\frac{0.05}{2}}$ $\times$ $\left(\frac{6}{\sqrt{20}}\right)$
Confidence Interval = 16 $\pm$ 2.09302 $\times$ $\left(\frac{6}{\sqrt{20}}\right)$

So, the margin of error = $\pm$ 2.808

95% confidence interval from 13.1919 to 18.8081

95% confidence interval from 13.1919 to 18.8081

2. ### Based on some survey, it is found that 195 of 250 randomly selected internet users have high-speed Internet access at home. Construct a 90% confidence interval for that proportion of all Web users who have high-speed Internet access at home.

Step 1 :

Given Sample Size n = 250

Frequency (x) = 195

Step 2 :

Formula Used Confidence Interval = p $\pm$ Z$_{\frac{\alpha}{2}} * \sqrt{\frac{p * q} {n} }(x, n - x \geq 5)$

Where,  p = x/n

q = 1- p

α = 1 - (Confidence Level / 100)

Zα/2 = Z-table value

Now p = $\frac{1 9 5}{250}$

= 0.78

q = 1- 0.78 = 0.22

$\alpha$ = 1 - ($\frac{90}{ 100}$)

= 1 - 0.9 = 0.1

On substitution in the formula we get the solution as

(0.729 < P < 0.831)